Hypercubes, Hyperplanes, and Constraint-Induced Complexity Collapse in Atomic Concept Learning

Abstract:We revisit higher-arity atomic concept learning through the geometry of hypercubes and hyperplanes of ground instances. Our starting point is the observation that the ambient r-dimensional hypercube of ground atoms is not structurally uniform. Its logical complexity is organized by hyperplanes: every hyperplane other than the full diagonal collapses into finitely many elementary-equivalence classes, with a bound independent of the term depth, while the full diagonal is exceptional and its class count grows without bound. This asymmetry is not merely geometric. It reflects the reduction-theoretic structure of the concepts themselves. Building on a higher-dimensional framework developed in the author's earlier work, we reinterpret these results through canonical simple concepts, minimal orderings, and representative reductions. This yields a taxonomy of hyperplane behavior in higher dimensions and shows that complexity is localized rather than spread uniformly through the instance space. The paper includes a fully worked binary case, an explicit treatment of the ternary hypercube, and an unpacked account of the reduction machinery that drives the collapse. The three-dimensional case already exhibits the essential phenomenon of orthogonal families, partial diagonals, and the exceptional full diagonal. This geometric-logical perspective clarifies where complexity is concentrated in atomic concept learning and suggests a modern interpretation in terms of constrained hypothesis spaces and structured classification.

Artificial Intelligence (cs.AI); Computation and Language (cs.CL); Logic in Computer Science (cs.LO)

arXiv:2608.02930 [cs.AI]

(or arXiv:2608.02930v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2608.02930 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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